THE ADIABATIC THEOREM AND LINEAR RESPONSE THEORY FOR EXTENDED QUANTUM SYSTEMS SVEN BACHMANN, WOJCIECH DE ROECK, AND MARTIN FRAAS Abstract. The adiabatic theorem refers to a setup where an evolution equation contains a time- dependent parameter whose change is very slow, measured by a vanishing parameter . Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instanta- neous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One im- portant application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g. for the integer quantum Hall effect. 1. Introduction 1.1. Adiabatic Theorems. The adiabatic theorem stands for a rather broad principle that can be described as follows. Consider an equation giving the evolution in time t of some '(t) 2 B, with B some Banach space: d (1.1) '(t) = L('(t); α ); dt t where L : B × R ! B is a smooth function and αt is a parameter depending parametrically on time. Now, assume that for αt = α frozen in time, the equation exhibits some tendency towards a fixpoint 1 R T 'α, as expressed, for example, by convergence in Cesaro mean T 0 '(t)dt ! 'α, as T ! 1 for any initial '(0). Intuitively, one can then expect that for a very slowly t-dependent αt, the solution '(t) shadows the instantaneous fixpoint 'αt if the initial condition '(0) is close to 'α0 . There is a multitude of results proving such a principle, starting from the old works [1, 2, 3], see [4, 5, 6, 7] for more recent accounts focusing on quantum dynamics, [8] for a version outside the quantum formalism, [9] for a version tailor-made for Markov processes, and [10] for a case where Lα(·) := L(·; α): B ! B is not linear. In order to guide the discussion, let us state explicitly the very general result of [11] that is however restricted to a linear Lα. Introducing a scaling parameter and a rescaled time s = t, we use s 2 [0; 1] as the basic variable and an s-dependent parameter αs, setting φ(s) = '(t). The above equation (1.1) then turns into arXiv:1705.02838v4 [math-ph] 27 Sep 2017 d (1.2) φ(s) = L φ(s): ds αs In [11], Lα is assumed to generate a contractive semigroup for any α, and to be such that B = KerLα ⊕RanLα. This corresponds to the intuition above, insofar as the latter condition is equivalent to mean-ergodicity Z T 1 tLα (1.3) e φ dt −! Pαφ, T 0 Date: September 29, 2017. 1 2 SVEN BACHMANN, WOJCIECH DE ROECK, AND MARTIN FRAAS for all φ 2 B, where Pα the ergodic projection on KerLα. The result of [11] is then that (1.4) sup k(1 − Pαs )φ(s)k ≤ C s2[0;1] where φ(s) is a solution of (1.2), with an initial condition φ(0) satisfying the same bound. In other words: the dynamically evolved φ(s) remains within the instantaneous kernel, up to a small leak. 1.2. Fundamental applications. There are at least two domains in physics where the adiabatic principle is fundamental. Thermodynamic Processes. We consider a phase space Ω (generalised positions and momenta) of a physical system, endowed with a symplectic structure and a Liouville measure !. A Hamilton function Hα :Ω ! R generates a flow on Ω, for which the measure ! is invariant. Let us choose B to be the class of signed measures that are absolutely continuous w.r.t. ! and we identify these measures with their densities ', so we can put B = L1(Ω;!). Then the flow on Ω is naturally lifted to B as d '(t) = fH ;'(t)g dt α where {·; ·} is the Poisson bracket corresponding to the symplectic structure. If the flow generated by the Hamiltonian Hα is ergodic on any energy shell ΩE = fx 2 Ω: H(x) = Eg and some technical conditions are met, then (1.3) is satisfied with Pα projecting on densities that are functions of Hα. Rescaling the time-dependent equation for the family Hαt to get d φ(s) = fH ; φ(s)g; ds αs we are in the framework described above and the result (1.4) applies. It teaches us that the evolution proceeds via instantaneous fixpoints, i.e. densities that are functions of Hα. Indeed, it is a basic tenet of thermodynamics that the evolution corresponding to a sufficiently slow αt | called in this context a quasi-static process | remains within the set of Gibbs states. −βHα (1.5) 'α,β / e ; where 0 < β < 1. Of course, the fixpoints are not automatically of the form (1.5) but typicality considerations involving `equivalence of ensembles' make them physically equivalent to Gibbs states, in the case where the system has a large number of degrees of freedom (see also below). Therefore, the adiabatic principle is often invoked to justify the relevance and prevalence of Gibbs state to describe time-inhomogenous systems. Quantum Dynamics. Here the Banach space B is a separable Hilbert space H, and there is a family of self-adjoint operators Hα (`Hamiltonians') acting on H. The dynamics of φ 2 H in rescaled time is given by the driven Schr¨odingerequation d (1.6) φ(s) = −iH φ(s): ds αs In this case, the condition (1.3) is satisfied by the spectral theorem. To exclude uninteresting trivial cases, we assume that KerHα is non-empty for all α. For simplicity, we also assume that Hα are bounded. Then, the result is given again by (1.4), which corresponds in this case to the original works of Born, Fock and Kato [1, 2, 3]. Note that despite the formulation, there is no distinguished role played by the eigenvalue 0 of Hα. Indeed, assume that Hα has an eigenvalue Eα, depending smoothly on α (recall that we assume everything to be smooth throughout). If φ(s) solves (1.6), then −(i/) R s E du φe(s) = e 0 αu φe(s) THE ADIABATIC THEOREM AND LINEAR RESPONSE THEORY FOR EXTENDED QUANTUM SYSTEMS 3 solves d (1.7) φe(s) = −iHeα φe(s); ds s with Heα = Hα −Eα. The latter equation (1.7) reduces the evolution problem for Eα-eigenvectors to the case above describing the evolution of vectors in the kernel. The adiabatic theorem in quantum mechanics is often invoked to argue that a system can be prepared in a certain state by switching the Hamiltonian slowly. Finally, it is worthwhile to note here that the adiabatic theorem can be substantially strengthened if the Hα have a spectral gap, i.e. if the spectra σ(Hα) n f0g are bounded away from 0, uniformly in α. In that case, at times in which the first m-derivatives of the Hamiltonian vanish, the result (1.4) m holds with an improved error bound Cm , where the integer m depends on the smoothness in α 1 and it can be made arbitrarily large if α 7! Hα is C . 1.3. Extended systems. The two above applications of the adiabatic theorem | thermodynamic processes and quantum dynamics | refer to dynamical systems of physics. A key parameter of any such dynamical system is the number N of degrees of freedom it has. A related important notion is that of the spatial extension of the physical system and its locality. Indeed, there are mathematical results that become irrelevant when the number of degrees of freedom grows large. Take for example the KAM theorem for Hamiltonian dynamics, stating that small perturbations of integrable systems inherit the integrability, see e.g. [12] and references therein. How small the perturbation must be depends on N and it must be vanishing with growing N. It is therefore commonly believed that the KAM theorem has little or no bearing on large physical systems. On the face of it, the adiabatic theorem suffers from a similar hitch, as the constant C in (1.4) grows typically linearly with N. As we will explain below, this divergence is unavoidable as the adiabatic theorem with an N-independent C is shown to be wrong. Yet, a lot of physics applications do involve macroscopic systems, where N ∼ 1023. Indeed, thermodynamics always assumes that the dynamical system is macroscopic, namely that one should take the `thermodynamic limit' N ! 1 which justifies the notion of typicality or the applicability of laws of large numbers. In quantum dynamics, many considerations involving the adiabatic theorem deal with extended lattices of spins. We mention most notably the problem of the classification of gapped phases or the dynamical crossing of a quantum critical point, both notions being meaningless for small systems where N is fixed. Hence there is a fundamental and practical need to find a form of the adiabatic theorem that is applicable for arbitrarily large systems. This is precisely what we provide in the present article for gapped quantum spin systems. There are two ingredients that will allow us to obtain again (1.4) with a N-independent constant C for macroscopic systems. First and foremost, we will exploit the locality of the system: only nearby degrees of freedom interact directly, and the quantum dynamics exhibits a finite speed of propagation.